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Effect of white noise on the $$d$$-dimensional harmonic oscillator. (Effet d’un bruit blanc sur l’oscillateur harmonique de dimension $$d$$.) (French) Zbl 0904.60039
The author studies the $$d$$-dimensional isotropic harmonic oscillator perturbed by small in some sense random term $$dW_tx_t$$, where $$W_t$$ denotes the Brownian motion. He obtains explicit formulas for the limit of Lyapunov type exponents $$t^{-1}\ln E(\| x_t\|^2)$$ and almost sure limit of $$t^{-1}\ln(\| x_t\|^2+\| v_t\|^2)$$, when the small parameter tends to zero.
##### MSC:
 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60H30 Applications of stochastic analysis (to PDEs, etc.) 93E15 Stochastic stability in control theory
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